Limit Theorems for Network Data without Metric Structure
Wen Jiang, Yachen Wang, Zeqi Wu, Xingbai Xu
TL;DR
This work develops limit theorems for random variables on networks without assuming a Euclidean metric, by extending the functional dependence measure to network dependence. It establishes moment inequalities, a concentration inequality, a law of large numbers, and central limit theorems framed through the network FDM Δ_{p,q}(𝓒_n), and demonstrates their applicability to nonlinear spatial autoregressive models. The results are validated for SAR-type networks and extended to transformations and multivariate settings, enabling robust inference without metric structure, particularly in networks with small diameters. Overall, the paper broadens the theoretical toolbox for econometric analysis of complex networks and offers practical criteria to verify limiting results in a wide range of non-metric network data.
Abstract
This paper develops limit theorems for random variables with network dependence, without requiring that individuals in the network to be located in a Euclidean or metric space. This distinguishes our approach from most existing limit theorems in network econometrics, which are based on weak dependence concepts such as strong mixing, near-epoch dependence, and $ψ$-dependence. By relaxing the assumption of an underlying metric space, our theorems can be applied to a broader range of network data, including financial and social networks. To derive the limit theorems, we generalize the concept of functional dependence (also known as physical dependence) from time series to random variables with network dependence. Using this framework, we establish several inequalities, a law of large numbers, and central limit theorems. Furthermore, we verify the conditions for these limit theorems based on primitive assumptions for spatial autoregressive models, which are widely used in network data analysis.